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arxiv: 2606.27140 · v1 · pith:6BDSQJNNnew · submitted 2026-06-25 · 💻 cs.LG

fTNN: a tensor neural network for fractional PDEs

Qingkui Ma , Hehu Xie , Xiaobo Yin This is my paper

Pith reviewed 2026-06-26 05:07 UTC · model grok-4.3

classification 💻 cs.LG
keywords fractional PDEstensor neural networksfractional Laplaciandeterministic quadratureboundary singularitiesneural network methodsadvection-diffusion
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The pith

A tensor neural network with geometry-adapted quadrature solves fractional PDEs accurately by handling singularities deterministically.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents fTNN, a deterministic tensor neural network method for solving fractional partial differential equations on bounded domains, such as the fractional Poisson equation and time-dependent fractional advection-diffusion equation. It establishes a geometry-adapted integration split that divides the fractional Laplacian into a singular near-field, regular far-field, and analytical exterior parts, each integrated with tailored quadrature rules to create a fully deterministic framework. Boundary-singularity-aware trial functions are constructed to resolve low-regularity solutions, with strategies for selecting exponents based on singularity structure. For time-dependent problems, a spatiotemporally separable network is combined with alternating optimization. Numerical experiments demonstrate high accuracy and substantial improvements over fPINN and Monte Carlo methods, especially for strong boundary singularities and long-time simulations. A sympathetic reader would care if this provides a reliable way to compute solutions to these nonlocal equations that appear in many applications.

Core claim

We develop the fTNN as a deterministic tensor neural network subspace method for the fractional Laplacian on bounded domains. The geometry-adapted integration split with spatially dependent near-field radius decomposes the operator into singular near-field, regular interior far-field, and analytical exterior far-field contributions. These are integrated using Gauss-Jacobi quadrature for singular radial integrals, Gauss quadrature for regular ones, and deterministic angular quadrature. Boundary-singularity-aware trial functions enriched with explicit boundary features, along with automatic selection of leading exponent and loss evaluation from singularity structure, are used to resolve low-re

What carries the argument

Geometry-adapted integration split featuring a spatially dependent near-field radius that decomposes the fractional Laplacian into three integrable contributions, paired with boundary-singularity-aware trial functions.

Load-bearing premise

The geometry-adapted integration split with a spatially dependent near-field radius decomposes the fractional Laplacian such that the chosen quadrature rules integrate the contributions accurately without errors that dominate the solution accuracy.

What would settle it

Numerical experiments on a benchmark fractional Poisson problem with strong boundary singularity showing that the solution error does not decrease below Monte Carlo baselines or fails to improve with refined quadrature.

Figures

Figures reproduced from arXiv: 2606.27140 by Hehu Xie, Qingkui Ma, Xiaobo Yin.

Figure 1
Figure 1. Figure 1: Architecture of the spatiotemporally separable neural network (STSNN). Black arrows denote linear (or affine) transformations. Each blue arrow indicates that the ending node is the product of all starting nodes of the same color. The final output is obtained by summing the contributions from the red arrows. The neural network is built with d + 1 subnetworks, and each subnetwork is a continuous mapping from… view at source ↗
Figure 2
Figure 2. Figure 2: Discretization of the fractional Laplacian on 2D domains. The black curve denotes the boundary ∂Ω. For each interior point x (colored dots), a local ball Br0(x) (x) is defined. R 2 is split into three regions: (1) near-field from x to ∂Br0(x) (x), using N0 = 10 symmetric Gaussian directions; (2) interior far-field from ∂Br0(x) (x) to ∂Ω, with N1 = 32 uniform directions; and (3) exterior far-field from ∂Ω t… view at source ↗
Figure 3
Figure 3. Figure 3: Numerical results of fTNN to solve (4.1) with exact solution (4.2). 0 2 4 6 8 10 12 Training steps (102) 1e+00 1e-01 1e-02 1e-03 1e-04 1e-05 1e-06 1e-07 Relative L 2 error ( e L2) Adam L-BFGS II : (α, β1, β2) = (0.40, 0.20, 0.30) III: (α, β1, β2) = (0.50, 0.05, 0.10) IV: (α, β1, β2) = (1.90, 1.50, 1.80) 0 2 4 6 8 10 12 Training steps (102) 1e-01 1e-02 1e-03 1e-04 1e-05 1e-06 1e-07 Relative L 2 error ( e L2… view at source ↗
Figure 4
Figure 4. Figure 4: Relative L 2 errors of fTNN with BFE (left) and BRFE (right) strategies to solve (4.1) with exact solution (4.2). 18 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The average computational time per epoch of fTNN for solving (1.1) with direction resolution n1 on the unit square (left) and the unit cube (right). refining a deterministic angular discretization rather than from increasing the number of Monte Carlo samples as used in QE-MC-fPINN [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Solutions of fTNN to solve the 2D fPE with RHS function f = 1. We then use fTNN to solve the fPE (1.1) with the RHS function f = 1 (clearly, s = 0) and different α in two dimensional case. Since the exact solutions are not known, we show the numerical solutions in [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Loss curves of fTNN to solve the 2D fPE with RHS function f = 1. 4.1.3 FPEs on the unit ball We consider fPE (1.1) on the unit ball, for which the corresponding RHS function could be analytically derived for a given solution. For brevity, we present the two-dimensional case, while the extension to three dimensions is analogous. In two dimensional case, it holds that (−∆)α/2  1 − ∥x∥ 2 2 1+α/2 = 2α Γ(α/2 … view at source ↗
Figure 8
Figure 8. Figure 8: Relative L 2 test error curves of fPINN-based methods (left column) and fTNN (right column) for solving fPE with exact solution (4.4), α = 1.5. For solution (4.4) with α = 1.5, all five methods behave well, although fTNN achieves substantially lower errors than the other methods. For solution (4.4) with α = 1.9, there are more pronounced differences among the errors of different methods. To be specific, th… view at source ↗
Figure 9
Figure 9. Figure 9: Plots of fTNN to solve fPE (1.1) on the unit ball. Top: exact solution (4.4) with α = 1.5, numerical solution, and the absolute error. Bottom: exact solution (4.5) with α = 1.9, numerical solution, and the absolute error. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Relative L 2 test errors for solving time-space fractional ADE (γ = 0.5, α = 1.50). The relative L 2 test errors of different methods are shown in [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The loss curve and the relative L 2 test error of fTNN for solving time-space fractional ADE (long-time simulation, T = 150). The loss curves in [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
read the original abstract

We develop the fTNN, a deterministic tensor neural network subspace method for problems involving the fractional Laplacian on bounded domains, taking the fractional Poisson equation and time-dependent fractional advection-diffusion equation as typical representatives. The work employs a geometry-adapted integration split featuring a spatially dependent near-field radius, which decomposes the fractional Laplacian into three contributions: a singular near field, a regular interior far field, and an analytical exterior far field. Then the singular radial integrals are treated by Gauss-Jacobi quadrature, the regular radial integrals by Gauss quadrature, and the angular variables by deterministic angular quadrature, yielding a fully deterministic integration framework of the fractional Laplacian operator. To accurately resolve low-regularity solutions and the associated loss functional, we construct boundary-singularity-aware trial functions enriched with explicit boundary features, and propose two strategies for automatically selecting the leading exponent and evaluating the loss function from the singularity structure induced by the fractional operator, or jointly by the fractional operator and the source term. For time-dependent fractional PDEs, we design a spatiotemporally separable neural network that factorizes the time-space residual into a sum of low-dimensional temporal and spatial integrals, and we integrate this representation with an alternating neural network subspace optimization strategy for efficient training. Numerical experiments show that the proposed framework attains high accuracy on the tested benchmarks and improves substantially over existing fPINN and Monte Carlo baselines, particularly for problems with strong boundary singularities and long-time simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript presents fTNN, a tensor neural network approach for solving fractional partial differential equations involving the fractional Laplacian on bounded domains. Key components include a geometry-adapted integration split with spatially dependent near-field radius decomposing the operator into near-field (Gauss-Jacobi quadrature), interior far-field (Gauss quadrature), and exterior analytical parts; boundary-singularity-aware trial functions with strategies for selecting singularity exponents; and for time-dependent problems, a spatiotemporally separable neural network combined with alternating subspace optimization. The authors claim that this framework achieves high accuracy on benchmarks, outperforming fPINN and Monte Carlo methods, especially for problems with strong boundary singularities and long-time simulations.

Significance. If the numerical performance claims are substantiated, the work offers a deterministic alternative to stochastic methods for fractional PDEs, potentially improving accuracy and reproducibility in handling singular solutions. The explicit incorporation of boundary features and the separable structure for time-dependent cases are notable technical contributions that could influence subsequent developments in neural methods for nonlocal operators.

major comments (2)
  1. [Abstract and integration framework description] The central accuracy claim depends on the spatially dependent near-field radius ensuring that the interior far-field integrand is sufficiently regular (C^∞ or high-order) for accurate Gauss quadrature at every collocation point. No a-priori analysis, adaptive criterion, or numerical check confirming this regularity—particularly near the domain boundary where the radius decreases—is provided, raising the risk that quadrature errors dominate the loss and affect solution accuracy for singular problems.
  2. [Numerical experiments] The asserted high accuracy and substantial improvements over fPINN and Monte Carlo baselines are presented without error bars on the reported errors, convergence tables (e.g., vs. number of quadrature points or network parameters), or detailed descriptions of the benchmark problems, exact solutions, and datasets used. This absence prevents independent assessment of the performance claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the integration framework and numerical validation. We address each major comment below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and integration framework description] The central accuracy claim depends on the spatially dependent near-field radius ensuring that the interior far-field integrand is sufficiently regular (C^∞ or high-order) for accurate Gauss quadrature at every collocation point. No a-priori analysis, adaptive criterion, or numerical check confirming this regularity—particularly near the domain boundary where the radius decreases—is provided, raising the risk that quadrature errors dominate the loss and affect solution accuracy for singular problems.

    Authors: We agree that the manuscript lacks an explicit a-priori regularity analysis or adaptive criterion for the far-field integrand under the spatially dependent radius. While the split is constructed so that the far-field kernel remains integrable and the radius choice excludes the singularity, no formal proof or boundary-specific numerical verification of quadrature accuracy is included. In the revised manuscript we will add a short subsection with a brief theoretical argument based on the kernel decay and radius scaling, together with numerical checks (e.g., quadrature-error tables near the boundary) to confirm that the far-field contribution remains below the target tolerance. revision: yes

  2. Referee: [Numerical experiments] The asserted high accuracy and substantial improvements over fPINN and Monte Carlo baselines are presented without error bars on the reported errors, convergence tables (e.g., vs. number of quadrature points or network parameters), or detailed descriptions of the benchmark problems, exact solutions, and datasets used. This absence prevents independent assessment of the performance claims.

    Authors: We acknowledge that the current numerical section does not report error bars, systematic convergence tables with respect to quadrature points or network parameters, or sufficiently detailed benchmark descriptions. In the revision we will expand the experiments section to include these elements: error bars from repeated runs where stochastic elements are present, convergence tables versus quadrature resolution and network size, and complete specifications of the benchmark problems, exact solutions, and data-generation procedures. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction from standard quadrature and NN components is self-contained

full rationale

The paper presents fTNN as a deterministic tensor NN method built from a geometry-adapted integration split (singular near-field via Gauss-Jacobi, regular far-field via Gauss, exterior analytic), boundary-singularity-aware trial functions, and spatiotemporally separable networks with alternating optimization. Accuracy claims rest on numerical experiments against fPINN and Monte Carlo baselines rather than any prediction or uniqueness result that reduces by construction to fitted parameters or self-citations. No equations or steps in the provided description equate outputs to inputs via definition, renaming, or load-bearing self-reference; the derivation chain remains independent of the target accuracy metrics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard assumptions about the decomposition of the fractional Laplacian and the accuracy of chosen quadrature rules; no new physical entities are introduced.

axioms (2)
  • domain assumption The fractional Laplacian on bounded domains admits a decomposition into singular near-field, regular interior far-field, and analytical exterior far-field contributions via a spatially dependent near-field radius.
    This split is the foundation of the deterministic integration framework described in the abstract.
  • domain assumption Gauss-Jacobi quadrature for singular radial integrals, Gauss quadrature for regular radial integrals, and deterministic angular quadrature together produce an accurate fully deterministic representation of the fractional operator.
    The abstract states that this combination yields the integration framework used by fTNN.

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